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Γραμμική Συνάρτηση
Γραμμική Συνάρτησις Linear Function thumb|300px| [[Γραμμική Συνάρτηση ]] thumb|300px| [[Γραμμική Συνάρτηση ]] thumb|300px| [[Γραμμική Συνάρτηση ]] thumb|300px| [[Μαθηματική Ανάλυση Συνάρτηση ---- Πεδίο Ορισμού Πεδίο Τιμών ---- Ενάρτηση Εφάρτηση Αμφάρτηση ---- Συναρτησιακή Μονοτονία Συναρτησιακή Συνέχεια Συναρτησιακή Σύγκλιση ]] - Μία συνάρτηση Ετυμολογία Η ονομασία "γραμμική" σχετίζεται ετυμολογικά με την λέξη "γραμμή". Ορισμός In mathematics, the term linear function refers to two distinct but related notions:"The term linear function, which is not used here, means a linear form in some textbooks and an affine function in others." Vaserstein 2006, p. 50-1 * In calculus and related areas, a linear function is a polynomial function of degree zero or one, or is the zero polynomial.Stewart 2012, p. 23 * In linear algebra and functional analysis, a linear function is a linear map.Shores 2007, p. 71 Μια απεικόνιση μεταξύ δύο διανυσματικών χώρων είναι γραμμική αν διατηρεί την δομή του διανυσματικού χώρου, δηλαδή μετασχηματίζει * το άθροισμα δύο διανυσμάτων σε άθροισμα των εικόνων τους και * το βαθμωτό πολλαπλάσιο ενός διανύσματος στο ίδιο βαθμωτό πολλαπλάσιο της εικόνας του. Εισαγωγή Διάκριση με τα Συναρτησιοειδή A Functional is a rule like a function but * the functional acts on a vector and maps it on to a scalar (we can say it "eats" a vector and produce a scalar), while *A Function is acting on an element of a set and maps it on other element of the same set. As a polynomial function In calculus, analytic geometry and related areas, a linear function is a polynomial of degree one or less, including the zero polynomial (the latter not being considered to have degree zero). When the function is of only one variable, it is of the form : f(x)=ax+b, where a'' and ''b are constants, often real numbers. The graph of such a function of one variable is a nonvertical line. a'' is frequently referred to as the slope of the line, and ''b as the intercept. For a function f(x_1, \ldots, x_k) of any finite number of independent variables, the general formula is : f(x_1, \ldots, x_k) = b + a_1 x_1 + \ldots + a_k x_k , and the graph is a hyperplane of dimension k''. A constant function is also considered linear in this context, as it is a polynomial of degree zero or is the zero polynomial. Its graph, when there is only one independent variable, is a horizontal line. In this context, the other meaning (a linear map) may be referred to as a homogeneous linear function or a linear form. In the context of linear algebra, this meaning (polynomial functions of degree 0 or 1) is a special kind of affine map. As a linear map In linear algebra, a linear function is a map ''f between two vector spaces that preserves vector addition and scalar multiplication: : f(\mathbf{x} + \mathbf{y}) = f(\mathbf{x}) + f(\mathbf{y}) : f(a\mathbf{x}) = af(\mathbf{x}). Here a'' denotes a constant belonging to some field ''K of scalars (for example, the real numbers) and x''' and '''y are elements of a vector space, which might be K'' itself. Some authors use "linear function" only for linear maps that take values in the scalar field;Gelfand 1961 these are also called linear functionals. The "linear functions" of calculus qualify as "linear maps" when (and only when) f(0,\ldots,0) = 0 , or, equivalently, when the constant b = 0 . Geometrically, the graph of the function must pass through the origin. Υποσημειώσεις Εσωτερική Αρθρογραφία * Μαθηματική Απεικόνιση * Πολυγραμμική Συνάρτηση * Γραμμική Μορφή * Homogeneous function * Nonlinear system * Piecewise linear function * Linear interpolation * Discontinuous linear map Βιβλιογραφία * Izrail Moiseevich Gelfand (1961), ''Lectures on Linear Algebra, Interscience Publishers, Inc., New York. Reprinted by Dover, 1989. ISBN 0-486-66082-6 * Thomas S. Shores (2007), Applied Linear Algebra and Matrix Analysis, Undergraduate Texts in Mathematics, Springer. ISBN 0-387-33195-6 *James Stewart (2012), Calculus: Early Transcendentals, edition 7E, Brooks/Cole. ISBN 978-0-538-49790-9 * Leonid N. Vaserstein (2006), "Linear Programming", in Leslie Hogben, ed., Handbook of Linear Algebra, Discrete Mathematics and Its Applications, Chapman and Hall/CRC, chap. 50. ISBN 1-584-88510-6 Ιστογραφία *Ομώνυμο άρθρο στην Βικιπαίδεια *Ομώνυμο άρθρο στην Livepedia * Γραμμικές Απεικονίσεις Κατηγορία:Συναρτήσεις Κατηγορία:Γραμμική Άλγεβρα